Once the working hypothesis is established, that the astronomical tidal function for any given port is a sum of a certain number of constituents whose frequencies are known a priori then the amplitudes and phases of the constituents may be determined by Fourier analysis.
To put the sum in more standard form, a constituent
Hcos(vt + phi)
will be rewritten as Acosvt + Bsinvt
(with A = Hcos(phi) and B = -Hsin(phi) as
usual).
The fundamental trigonometric identities that make Fourier analysis work imply that for different speeds v and w
/T
|
(1/T)| cos(vt) cos(wt) dt ---> 0
|
/0
as T --> infty
and similarly for the products cos(vt) sin(wt), cos(vt) sin(vt), and sin(vt) sin(wt), whereas
/T
|
(1/T)| cos(vt) cos(vt) dt ---> 1/2
|
/0
as T --> infty
and the same for
/T
|
(1/T)| sin(vt) sin(vt) dt
|
/0
So if R(t) is the record from a tide gauge at the port
in question, the cosine and sine amplitudes A and B
for a particular speed v
may be determined by
/T
|
A = (2/T)| R(t) cos(vt) dt
|
/0
/T
|
B = (2/T)| R(t) sin(vt) dt
|
/0
for sufficiently large T.
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